The multilinear fractional sparse operator theory I: pointwise domination and weighted estimate

How to establish some specific quantitative weighted estimates for the generalized commutator of multilinear fractional singular integral operator $\mathcal{T}_{\eta}^{{\bf b}}$ is the focus of this paper, which is defined by $$\mathcal{T}_{\eta}^{{\bf b}}(\vec{f})(x):= \mathcal{T}_{\eta}\left((b_1(x) - b_1)^{\beta_1}f_1,\ldots,(b_m(x) - b_m)^{\beta_m}f_m\right)(x),$$ where $\mathcal{T}_{\eta}$ is a multilinear fractional singular integral operator, ${\bf b}:=({b_1}, \cdots ,{b_m})$ is a set of symbol functions, and $({\beta_1}, \cdots ,{\beta_m}) \in {\mathbb{N}_0^m}$. Pointwise dominating the aforementioned commutator leads us to consider a class of higher order multi-symbol multilinear fractional sparse operator ${\mathcal A}_{\eta ,\mathcal{S},\tau}^\mathbf{b,k,t}$ to achieve this long-cherished wish. Therefore, it suffices to construct its quantitative weighted estimates, which firstly include the characterization of several types of multilinear weighted conditions $A_{\vec p,q}^*$, $W_{\vec p,q}^\infty$, and $H_{\vec p,q}^\infty$. Within the scope of this work, Bloom type estimate for first order multi-symbol multilinear fractional sparse operator is established herein. Moreover, we derive two distinct Bloom type estimates for higher order multi-symbol multilinear fractional sparse operator by using "maximal weight method" and "iterated weight method" respectively, which not only refines some of Lerner's methods but greatly enhances the generality of our conclusions. Endpoint quantitative estimates for multilinear fractional singular integral operators and their first order commutators are also obtained as the last main result. It is also worthy of highlighting that some important multilinear fractional operators are applicable to our results as applications.